Advertisement

Monatshefte für Mathematik

, Volume 179, Issue 2, pp 191–199 | Cite as

Invariants of domains with finite GK dimension

  • Jeffrey BergenEmail author
  • Piotr Grzeszczuk
Article
  • 123 Downloads

Abstract

This paper examines the invariants of automorphisms, derivations, and \(q\)-skew derivations of finitely generated domains with finite GK dimension. We generalize results on centralizers obtained by Bell and Small. We show that if a \(q\)-skew derivation is not algebraic, then the GK dimension of the domain exceeds that of its invariants by at least one. In addition, if the domain has GK dimension less than 3 and the ground field is algebraically closed, then the invariants must be commutative.

Keywords

GK dimension Invariants Automorphisms Derivations  Skew derivations 

Mathematics Subject Classification

16P90 16W25 

References

  1. 1.
    Bell, J.P.: Centralizers in domains of finite Gelfand–Kirillov dimension. Bull. Lond. Math. Soc. 41(3), 559–562 (2009)zbMATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    Bell, J.P., Small, L.W.: Centralizers in domains of Gelfand–Kirillov dimension 2. Bull. Lond. Math. Soc. 36(6), 779–785 (2004)zbMATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    Bergman, G.M.: A note on growth functions of algebras and semigroups. In: Research Note. University of California, Berkeley (1978). (Unpublished mimeographed notes)Google Scholar
  4. 4.
    Borho, W., Kraft, H.: Über die Gelfand–Kirillov dimension. Math. Ann. 220, 1–24 (1976)zbMATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    Chuang, C.-L., Lee, T.-K.: \(q\)-skew derivations and polynomial identities. Manuscripta Math. 116, 229–243 (2005)zbMATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    Kharchenko, V.K.: Centralizers in prime rings. Algebra Log. 20(2), 157–167 (1981)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Kharchenko, V.K.: Shirshov finiteness over rings of constants. Algebra Log. 36(2), 219–238 (1997)zbMATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    Krause, G.R., Lenagan, T.H.: Growth of algebras and Gelfand–Kirillov dimension. In: Graduate Studies in Mathematics, vol. 22, Am. Math. Soc., Providence (2000)Google Scholar
  9. 9.
    Leroy, A., Matczuk, J.: Dérivations et automorphismes algébriques d’anneaux premiers. Commun. Algebra 13(6), 1245–1266 (1985)zbMATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    McConnell, J.C., Robson, J.C.: Noncommutative noetherian rings. In: Pure and Applied Mathematics, John Wiley & Sons, Chichester (1987)Google Scholar
  11. 11.
    Montgomery, S.: \(X\)-inner automorphisms of filtered algebras. Proc. Am. Math. Soc. 83(2), 263–268 (1981)zbMATHGoogle Scholar
  12. 12.
    Small, L.W., Warfield R.B. Jr: Prime affine algebras of Gelfand–Kirillov dimension one. J. Algebra 91(2), 386–389 (1984)Google Scholar
  13. 13.
    Zhang, J.J.: On Gelfand–Kirillov transcendence degree. Trans. Am. Math. Soc. 348(7), 2867–2899 (1996)zbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Wien 2015

Authors and Affiliations

  1. 1.Department of MathematicsDePaul UniversityChicagoUSA
  2. 2.Faculty of Computer ScienceBiałystok University of TechnologyBiałystokPoland

Personalised recommendations